Alan

K/M = 2/5     so  K = 2M/5

M = K + 105   so  M = 2M/5 + 105   or 3M/5 = 105  so M = 175

This means K = 175 - 105 = 70

Hence L = 385 - 175 - 70  or L = 140

As a percentage of 385, Laura gets 100L/385 = 100*140/385 → 36.36%

An algorithm is a step by step procedure for getting from a starting point to an end point. It is to mathematics what a recipe is to baking!

f(x) is an odd function (i.e. f(-x) = - f(x)) and its derivatives are alternately even and odd.  This means that if you were to count f(x) as f⁽¹⁾(x) then f⁽²⁰¹⁷⁾(x) would also be odd and the integral would be zero.

However, if you really mean f⁽²⁰¹⁷⁾(x) is the 2017th derivative of f(x) then it is an even function and I don't see a simple way of determining it (that's not to say there isn't a way; just that I can't see it!).

When you get a question with "at least" in, it is often a good idea to find the probability of "none" first, then the probability of "at least" = 1 - probability of "none".

Suppose there are n integers between 1 and 50 that are not multiples of 3.  Then the probability of choosing a random integer that is not a multiple of 3 is n/50.

The probability of doinjg this twice is (n/50)^2.  Hence the probability that at least one number is a multiple of 3 is 1 - (n/50)^2

asinus's graphical roots suggest that (2x-1) and (x-3) are also factors, so you could try expanding

(2x-1)(x-3)(x-a)(x-b) and comparing the resulting coefficients with  2x^4-7x^3+5x^2-7x+3 to find a and b.

I suspect your first expression should be:  2x^5-9x^4+12x^3-12x^2+10x-3  as x-1 is a factor of this, but isn't a factor of 2x^5-9x^4+12x^2+10x-3.

"The diagram showsthe cross-section of a cylindrical pipewith water lying in the bottom.

(a) If the maximum depth of the water is 2 cm and the radius of the pipe is 7 cm, find the area shaded.

(b) what is the volume of water in a length of 30 cm?"

(a)

(b) Simply multiply A3 by 30 cm.

This says i^i is a member of the set of Reals.  i^i = e^-(pi/2) which is a Real number.

Not exactly rib-ticklingly funny though!

"Irrational" means cannot be expressed as the ratio of two integers (I.e. whole numbers).  So you could never have a circumference divided by a diameter which could be represented by the ratio of two whole numbers.

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Perhaps you would be interested in quaternions Max (see https://en.wikipedia.org/wiki/Quaternion for example).  These extend the concept of complex numbers.